Determine if the following two planes 2x+3y-z=4 and 3x-4y-6z=1 are parallel, orthogonal or coincidental

To determine whether two planes are parallel, orthogonal (perpendicular), or coincidental (the same plane), we need to compare their normal vectors.

The equation of a plane in the form Ax + By + Cz = D can be written as a vector equation (A, B, C) · (x, y, z) = D. From this equation, we can extract the normal vector of the plane, which is given by (A, B, C).

For the first plane, 2x + 3y - z = 4, the normal vector is (2, 3, -1).
For the second plane, 3x - 4y - 6z = 1, the normal vector is (3, -4, -6).

To determine whether two planes are parallel or orthogonal, we can take the dot product of their normal vectors. If the dot product is zero, the planes are orthogonal, meaning they are perpendicular to each other. If the dot product is nonzero, the planes are parallel, meaning they have the same or proportional normal vectors.

Calculating the dot product of the two normal vectors:
(2, 3, -1) · (3, -4, -6) = (2 × 3) + (3 × -4) + (-1 × -6) = 6 - 12 + 6 = 0

Since the dot product of the normal vectors is zero, the two planes are orthogonal to each other.